Geometry with Trigonometry

10 Preliminaries Ch. 1

1.5 Revisionofgeometricalconcepts

1.5.1

As part of the preliminary programme, we now include a review of the basic concepts
of geometry. Geometry should be thought of as arising from an initial experimental
and observational stage, where the figures are looked at and there is a great emphasis
on a visual approach.

1.5.2 Thebasicshapes............................

1. The planeΠis a set, the elements of which are calledpoints. Certain subsets ofΠ
   are calledlines.

By observation, given any distinct pointsA,B∈Π, there is a unique line to which
AandBboth belong. It is denoted byAB.

2. Given distinct pointsAandB, the set of points consisting ofAandBthemselves
   and all the points of the lineABwhich are betweenAandBis called asegment,and
   denoted by[A,B].

A

B

Figure 1.1. A lineAB.

The arrows indicate that the line

is to be continued unendingly.

A

B

Figure 1.2. A segment[A,B].

NOTE. Note that the modern mathematical terminology differs significantly from that
inThe Elements. What was called a ‘line’ is now called a segment, and we have added
the new concept of ‘line’. This is confusing, but the practice is well established. In
ordinary English and in subjects cognate to mathematics, ‘line’ has its old meaning.

3. The set consisting of the pointAitself and all the points of the lineABwhich
   are on the same side ofAasBis, is called ahalf-line, and denoted by[A,B.IfAis
   betweenBandC, then the half-lines[A,Band[A,Care said to beopposite.